Wir schaffen Wissen heute für morgen NLO QCD corrections to Higgs boson pair production via gluon fusion Seraina Glaus Theory Group LTP, Paul Scherrer Institute & University Zurich In collaboration with M. Spira, M. Mühlleitner, J. Baglio, F. Campanario, J. Streicher
Outline Motivation Objective Previous work NLO Cross section - Virtual Corrections - Real Corrections Numerical Analysis Conclusions/Outlook
Motivation Detection of a Higgs boson with a mass ~ 125 GeV
Motivation Further investigations of the properties of the detected particle for a unique association to a model Higgs mass, coupling strengths, spin and CP already determined self-coupling strength still unknown 2
Motivation Higgs boson pair production Production channel Cross section Baglio, Djouadi, Gröber, Mühlleitner, Quevillon, Spira
Motivation Uncertainties: Baglio, Djouadi, Gröber, Mühlleitner, Quevillon, Spira
Objective Gluon fusion gg HH: loop induced Complete calculation of the NLO QCD corrections (2-loop) considering the top- und bottom mass dependences in the context of the Standard Model 2-loop integrals with 3 kinematical parameter ratios
Previous work Virtual & real (N)NLO QCD corrections in large top mass limit: ~100% Large top mass expansion: ~ ±10% Dawson,Dittmaier,Spira de Florian,Mazzitelli Grigo,Melnikov,Steinhauser NLO mass effects of the real NLO correction alone ~ -10 % NLO QCD corrections including the full top mass dependence Grigo, Hoff, Melnikov, Steinhauser Frederix, Frixione, Hirschi, Maltoni, Mattelaer, Torrielli, Vryonidou, Zaro Borowka, Greiner, Heinrich, Jones, Kerner, Schlenk, Schubert, Zirke
NLO Corrections σ LO : Δσ virt : Δσ ij :
NLO Corrections
Virtual Corrections Set-up - Associate the 47 two-loop box diagrams to similar topologies - Generate matrix elements for all possible diagrams (by hand Reduce, Mathematica) - Use dimensional regularisation: n = 4 2ε - Perform Feynman parametrisation 6-dimensional integrals
Virtual Corrections Divergences - Extraction of the ultraviolet divergences of the matrix elements using endpoint subtractions of the 6-dimensional Feynman integrals - Extraction of the infrared and collinear divergences using a proper subtraction of the integrand denominator: Z 1 0 d~xdr N = ar 2 + br + c N 0 = br + c rh(~x, r) N 3+2 = Z 1 0 a = O( ) b =1+O( ) c = s x(1 x)(1 s)t n rh(~x, r) rh(~x, 0) d~xdr N 3+2 N0 3+2 + s = rh(~x, 0) o N0 3+2 ŝ m 2 Q Taylor expansion in analytical r-integration
Virtual Corrections Divergences - Integration by parts due to numerical instabilities at the thresholds M 2 HH > 4m 2 Q ) m 2 Q! m 2 Q(1 i ) with 1 Z 1 0 dx f(x) (a + bx) 3 = f(0) 2a 2 Z f(1) 1 2(a + b) 2 + 0 f 0 (x) dx 2(a + bx) 2 - gluon rescattering: threshold for M 2 HH > 0
Virtual Corrections Renormalisation αs and mq need to be renormalised αs in MS with N F = 5 mq on shell = s LO s + m t LO m t Subtraction of the heavy-top limit C mass = C 0 C 0 HTL virtual mass effects only (infrared finite) Adding back the results of HPAIR (heavy-top limit) C = C HTL + C mass HPAIR
Virtual Corrections Remaining Diagrams Triangular diagrams single Higgs case One-particle reducible diagrams analytical results (H! Z ) see e.g. Degrassi, Giardino, Gröber
Real Corrections Processes: gg,qq HHg; gq HHq Full matrix elements generated with LoopTools Matrix elements in the heavy-top limit rescaled locally by massive LO matrix elements (with adjusted kinematics) subtracted free of divergences Adding back the results of HPAIR (heavy-top limit)
Numerical Analysis Use Vegas for numerical integration P. Lepage d Calculation of differential cross section dq 2, (Q 2 = m 2 HH) Q 2 d virt dq 2 = dlgg d ˆvirt(Q 2 ) = Q 2 s partonic cross section (seven-dimensional integrals) Thresholds: m 2 Q! m 2 Q(1 i ) different Richardson extrapolation (NWA) Integration of real corrections straight forward =0
Conclusions and Outlook Full NLO calculation close to finalisation numerical results soon NLO mass effects expected in the 10-20 % range Independent cross check of existing results Outlook: Extension of the calculation to BSM-Higgs scenarios (dim 6, 2HDM)
Back-up Richardson Extrapolation M 2 [f(h),f(2h)] = 2f(h) f(2h) M 4 [f(h),f(2h),f(4h)] = (8f(h) 6f(2h)+f(4h))/3 M 8 [f(h),f(2h),f(4h),f(8h)] = (64f(h) 56f(2h) + 14f(4h) f(8h))/21 f(x) f(x) polynomial for small h h 2h 4h 8h x